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# Binomial distributions

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### Binomial distributions

1. 1. Binomial Distribution
2. 2. 4 Parts of a Binomial Distribution 1. Outcomes are success or not success 2. Probability of Success is Fixed 3. Fixed number of trials 4. Trials are Independent Notation: B(π,π) Mean of Binomial Distribution: Number of trials * Probability of Success E(X)= π = ππ Standard Deviation of Binomial Distribution: π = ππ(1 β π) n= number of trials p=probability of success
3. 3. Probability of a Binomial Event Occurring k times π P(X=k) = ( π )π π (1 β π) πβπ Ex. The probability the team will win is 40%. What is the probability the team will win 6 of its next 8 games? Identify n,p,k n= number of trials (games) = 8 p = probability of success (winning) = 0.4 k= # of times event occurs (# wins) = 6 Apply to formula: π P(X=k) = ( π )π π (1 β π) πβπ P(X=6) = (8 )0.46 (1 β 0.4)8β6 = 0.0412 6 You may use the Binomial Pdf function on your calculator for this.
4. 4. Probability of a Binomial Event occurring at least or at most k times This is done the same way as for exactly, except we need to do this calculation multiple times and then add them up. Ex. The probability the team will win is 40%. What is the probability the team will win at least 6 of its next 8 games? Identify n, p, and k n=8 p = 0.4 k = 6,7,and 8 (at least means include 6 and higher) Calculation: π P(X=k) = ( π )π π (1 β π) πβπ P(Xβ₯6) = (8 )0.46 (1 β 0.4)8β6 +(8 )0.47 (1 β 0.4)8β7 +(8 )0.48 (1 β 0.4)8β8 7 6 8 = 0.041288 + 0.007864 + 0.000655 = 0.049807 =0.050 You may also use the Binomial Cdf function on your calculator If the questions asks for at most you do all of the values at or below the given value For questions asking less than or more than, do not include the given value